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ON THE MOTION  OF PENDULUMS.
51
and if the logarithms of the coefficients of ttT1, ru~2..
be required, it will be sufficient to add "1505150 to the 1st, 3rd,
5th, &c. of the logarithms above given.
36. It will be found that when ttt is at all large, the series (113) are at first convergent, and afterwards divergent, and in passing from convergent to divergent the quantities ut become nearly equal for several successive terms. If after having calculated i terms of the first of the series (113) we wish to complete the series by a formula involving the differences of ut, we have
and
= 2 cos
.
+ I 4 SOC 7T      COS         Q        7T .
o/          o
 1                      t            7T    .                     A
t)     TT . w. - i sec   sin -0 TT . A/r.
o                          o        o
so that the quantities to be added to k, 7c, arc
^   7     /    TV* i       TT f     2i  1              -       TT      2i      A
t() Kt     ( 1)   o SCC .. -COS               7T . U,  J SCC -r- COS -  7T . Aw/,
o (          o                         o        o
,       ,,       ,      1W ,
to A: ,   ( 1.) i see .
*i7. rrink following tn,blo oont.;uns tlio values of the functions /.; ;i.nd /;' calctula-tcd for 40 diftorcnl, values of M. From ltt = 'l to 1U--'I'5 UK; cahuibition WJIK performed by me;ius of tlie formula, (105) ; the rest of Uie table was calculated by means of the series (113). In the former part of the calculation, six places of decimals were employed in calculating the functions Jl/0, &e. given by (KW). Thii la,st iigunj wa,s then struc.k out, and five-figure logarithms wen; employed in multiplying the four functions 7lfu,
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